Universite Laval, Quebec July 9 – 13, 2018

Canadian Number Theory Association ConferenceCNTA XV

Generalization of the Landau–Ramanujan

constant for sums of two squaresto other binary forms

Michel Waldschmidt

Institut de Mathematiques de Jussieu — Paris VIhttp://webusers.imj-prg.fr/~michel.waldschmidt/

update: 03/07/2018

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Abstract

Let F 2 Z[X, Y ] be a binary form with integer coe�cients,non-zero discriminant, and degree d. As N ! 1, the numberof integers m 2 Z, |m| N which are representable by F isasymptotically• CFN(logN)�

12 if d = 2 and F is a positive definite

quadratic form (P. Bernays),and• CFN

2d if d � 3 (C.L. Stewart – S.Y. Xiao).

For the cyclotomic quadratic form �4(X, Y ) = X2 + Y

2, theconstant C�4 is the so–called Landau–Ramanujan constant.

We consider cyclotomic binary forms, which are thehomogeneous versions of the cyclotomic polynomials.

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This is a joint work with Etienne Fouvry andClaude Levesque

Etienne Fouvry Claude Levesque

Representation of integers by cyclotomic binary forms.

Acta Arithmetica, 20p. Online First March 2018.Dedicated to Rob Tijdeman. arXiv: 712.09019 [math.NT]

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The Landau–Ramanujan constant

Edmund Landau1877 – 1938

Srinivasa Ramanujan1887 – 1920

The number of positive integers N which are sums of twosquares is asymptotically C�4N(logN)�

12 , where

C�4 =1

212

·Y

p⌘ 3 mod 4

✓1� 1

p2

◆� 12

.

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Online Encyclopedia of Integer Sequenceshttps://oeis.org/A064533

[OEIS A064533] Decimal expansion ofLandau-Ramanujan constant.

C�4 = 0.764 223 653 589 220 . . .

• Ph. Flajolet and I. Vardi, Zeta function expansions of someclassical constants, Feb 18 1996.• Xavier Gourdon and Pascal Sebah, Constants and records ofcomputation.• David E. G. Hare, 125 079 digits of the Landau-Ramanujanconstant.

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The Landau–Ramanujan constant

References : https://oeis.org/A064533

• B. C. Berndt, Ramanujan’s notebook part IV,Springer-Verlag, 1994• S. R. Finch, Mathematical Constants, Cambridge, 2003, pp.98-104.• G. H. Hardy, ”Ramanujan, Twelve lectures on subjectssuggested by his life and work”, Chelsea, 1940.• Institute of Physics, Constants - Landau-RamanujanConstant• Simon Plou↵e, Landau Ramanujan constant• Eric Weisstein’s World of Mathematics, Ramanujan constant• https://en.wikipedia.org/wiki/Landau-Ramanujan_constant

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Sums of two squares

A prime number is a sum oftwo squares if and only if it iseither 2 or else congruent to 1modulo 4.

Pierre de Fermat1607 ( ?) – 1665

Identity of Brahmagupta :

(a2 + b2)(c2 + d

2) = e2 + f

2

with

e = ac� bd, f = ad+ bc. Brahmagupta598 – 668

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Sums of two squares

If a and q are two integers, we denote by Na,q any integer � 1satisfying the condition

p | Na,q =) p ⌘ a mod q.

An integer m � 1 can be written as

m = �4(x, y) = x2 + y

2

if and only if there exist integers a � 0, N3,4 and N1,4 suchthat

m = 2a N23,4 N1,4.

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Positive definite quadratic formsLet F 2 Z[X, Y ] be a positive definite quadratic form. Thereexists a positive constant CF such that, for N ! 1, thenumber of positive integers m 2 Z, m N which arerepresented by F is asymptotically CFN(logN)�

12 .

Paul Bernays1888 – 1977

P. Bernays, Uber die

Darstellung von positiven,

ganzen Zahlen durch die

primitiven, binaren

quadratischen Formen einer

nicht quadratischen

Diskriminante, Ph.D.dissertation,Georg-August-Universitat,Gottingen, Germany, 1912.

http://www.ethlife.ethz.ch/archive_articles/120907_bernays_fm/9 / 47

Paul Bernays (1888 – 1977)https://www.thefamouspeople.com/profiles/paul-bernays-7244.php

• 1912, Ph.D. in mathematics, University of Gottingen, On the

analytic number theory of binary quadratic forms (Advisor :E. Landau).• 1913, Habilitation, University of Zurich, On complex analysis and

Picard’s theorem, advisor E. Zermelo.• 1912 – 1917, Zurich ; work with Georg Polya, A. Einstein,Hermann Weyl.• 1917 – 1933, Gottingen, with D. Hilbert. Studied with EmmyNoether, van der Waerden, G. Herglotz,• 1935 – 1936, Institute for Advanced Study, Princeton. Lectureson mathematical logic and axiomatic set theory• 1936 —, ETH Zurich.• With Hilbert, “Grundlagen der Mathematik” (1934 – 39) 2 vol.— Hilbert–Bernays paradox.• Axiomatic Set Theory (1958). —Von Neumann–Bernays–Godel set theory.

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Generalizations

• Sums of cubes, biquadrates,. . .

Notice that X3 + Y3 = (X + Y )(X2 �XY + Y

2)

We start with the quadratic form �3(X, Y ) = X2 +XY + Y

2

which is the homogeneous version of the cyclotomicpolynomial �3(t) = t

2 + t+ 1.Notice that

�6(X, Y ) = �3(X,�Y ) = X2 �XY + Y

2

Also�8(X, Y ) = X

4 + Y4.

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The quadratic form x2 + xy + y

2

A prime number is represented by the quadratic formx2 + xy + y

2 if and only if it is either 3 or else congruent to 1modulo 3.Product of two numbers represented by the quadratic formx2 + xy + y

2 :

(a2 + ab+ b2)(c2 + cd+ d

2) = e2 + ef + f

2

withe = ac� bd, f = ad+ bd+ bc.

The quadratic cyclotomic field Q(p�3) = Q(⇣3),

1 + ⇣3 + ⇣32 = 0 :

a2 + ab+ b

2 = NormQ(⇣3)/Q(a� ⇣3b).

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Loeschian numbers : m = x2 + xy + y

2

An integer m � 1 can be written as

m = �3(x, y) = �6(x,�y) = x2 + xy + y

2

if and only if there exist integers b � 0, N2,3 and N1,3 suchthat

m = 3b N22,3 N1,3.

The number of positive integers N which are represented bythe quadratic form x

2 + xy + y2 is asymptotically

C�3N(logN)�12 , where

C�3 =1

2123

14

·Y

p⌘ 2 mod 3

✓1� 1

p2

◆� 12

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Zeta function expansions of some classical constants, Feb 18 1996.

Philippe Flajolet Ilan Vardi

Bill Allombert

C�3 = 0.638909405445343882254942674928245093754975508029123345421692365708076310027649658246897179112528664388141687519107424 . . .

April 2018

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Online Encyclopedia of Integer Sequences OEISA301429

[OEIS A301429] Decimal expansion of an analog ofthe Landau-Ramanujan constant for Loeschiannumbers.

The first decimal digits of C�3 are

C�3 = 0.638 909 405 44 . . .

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Loeschian numbers which are sums of two squaresAn integer m � 1 is simultaneously of the forms

m = �4(x, y) = x2 + y

2 and m = �3(u, v) = u2 + uv + v

2

if and only if there exist integers a, b � 0, N5,12, N7,12, N11,12

and N1,12 such that

m =⇣2a 3b N5,12 N7,12 N11,12

⌘2N1,12.

The number of Loeschian integers N which are sums of twosquares is asymptotically �N(logN)�3/4, where

� =3

14

254

·⇡ 12 ·(log(2+

p3))

14 · 1

�(1/4)·

Y

p⌘ 5, 7, 11 mod 12

⇣1� 1

p2

⌘� 12.

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OEIS A301430 � = 0.302 316 142 35 . . .[ OEIS A301430] Decimal expansion of an analog ofthe Landau-Ramanujan constant for Loeschiannumbers which are sums of two squares.

� =3

14

254

·⇡ 12 ·(log(2+

p3))

14 · 1

�(1/4)·

Y

p⌘ 5, 7, 11 mod 12

⇣1� 1

p2

⌘� 12

Bill Allombert

� = 0.3023161423570656379477699004801997156024127951893696454588678412888654487524105108994874678139792727085677659132725910666837135863 . . .

April 2018

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Numbers represented by a cyclotomic binary formof degree � 2

For N � 1, the number of m N for which there existsn � 3 and (x, y) 2 Z2 with max(|x|, |y|) � 2 andm = �n(x, y), is asymptotically

(C�4 + C�3)N

(logN)12

� �N

(logN)34

+O

N

(logN)32

!

as N ! 1.

C�4 + C�3 = 1.403 133 059 034 . . . � = 0.302 316 142 35 . . .

Etienne Fouvry, Claude Levesque & M.W. ; Representation of

integers by cyclotomic binary forms. Acta Arithmetica, 20p.Online First March 2018.Dedicated to Rob Tijdeman. arXiv: 712.09019 [math.NT]

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Higher degreeThe situation for positive definite forms of degree � 3 isdi↵erent for several reasons.• If a positive integer m is represented by a positive definitequadratic form, it usually has many such representations ;while if a positive integer m is represented by a positivedefinite binary form of degree d � 3, it usually has few suchrepresentations.

• If F is a positive definite quadratic form, the number of(x, y) with F (x, y) N is asymptotically a constant times N ,but the number of F (x, y) is much smaller.

• If F is a positive definite binary form of degree d � 3, thenumber of (x, y) with F (x, y) N is asymptotically aconstant times N

2d , the number of F (x, y) is also

asymptotically a constant times N2d .

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Higher degree

A quadratic form has infinitely many automorphisms, a binaryform of higher degree has a finite group of automorphisms.

Stanley Yao Xiao

S. Yao Xiao, On the representation of integers by binary

quadratic forms.arXiv:1704.00221

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Sums of k–th powers

If a positive integer m is a sum of two squares, there are manysuch representations.Indeed, the number of (x, y) in Z⇥ Z with x

2 + y2 N is

asymptotic to ⇡N , while the number of values N taken bythe quadratic form �4 is asymptotic to C�4N/

plogN where

C�4 is the Landau–Ramanujan constant. Hence �4 takes eachof these values with a high multiplicity, on the average(⇡/C�4)

plogN .

On the opposite, given an integer k � 3, that a positiveinteger is a sum of two k–th powers in more than one way(not counting symmetries) is• rare for k = 3,• extremely rare for k = 4,• maybe impossible for k � 5.

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1729 : the taxicab number

The smallest positive integer which is sum of two cubes in twoessentially di↵erent ways :

1729 = 103 + 93 = 123 + 13.

Godfrey Harold Hardy1877–1947

Srinivasa Ramanujan1887 – 1920

1657 : Frenicle de Bessy (1605 ? – 1675)

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The sequence of Taxicab numbers

[OEIS A001235] Taxi-cab numbers: sums of 2 cubes inmore than 1 way.

1729 = 103+93 = 123+13, 4104 = 23+163 = 93+153, . . .

1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232,65728, 110656, 110808, 134379, 149389, 165464, 171288, 195841,216027, 216125, 262656, 314496, 320264, 327763, 373464, 402597,439101, 443889, 513000, 513856, 515375, 525824, 558441, 593047, . . .

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Another sequence of Taxicab numbers (Fermat)

[OEIS A011541] Hardy-Ramanujan numbers: thesmallest number that is the sum of 2 positiveintegral cubes in n ways.http://mathworld.wolfram.com/TaxicabNumber.htmlT a(1) = 2,

T a(2) = 1729 = 103 + 93 = 123 + 13,

T a(3) = 87 539 319 = 1673 + 4363 = 2283 + 4233 =2553 + 4143,

T a(4) = 6 963 472 309 248 = 24213 + 19 0833 =54363 + 18 9483 = 10 2003 + 18 0723 = 13 3223 + 16 6303,

T a(5) = 48 988 659 276 962 496,T a(6) = 24 153 319 581 254 312 065 344.

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Cubefree taxicab numbers

15 170 835 645 = 5173+24683 = 7093+24563 = 17333+21523.

The smallest cubefree taxicabnumber with threerepresentations was discoveredby Paul Vojta (unpublished)in 1981 while he was agraduate student.

Paul Vojta

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Cubefree taxicab numbers

Stuart Gascoigne and Duncan Moore (2003) :1 801 049 058 342 701 083 = 922273 + 12165003 = 1366353 +12161023 = 3419953 + 12076023 = 6002593 + 11658843

[OEIS A080642] Cubefree taxicab numbers: thesmallest cubefree number that is the sum of 2cubes in n ways.

https://en.wikipedia.org/wiki/Taxicab_number

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Taxicabs and Sums of Two Cubes

If the sequence (an) of cubefree taxicab numbers with n

representations is infinite, then the Mordell-Weil rank of theelliptic curve x

3 + y3 = an tends to infinity with n.

Joseph Silverman

J. H. Silverman, Taxicabs andSums of Two Cubes, Amer.Math. Monthly, 100 (1993),331-340.

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635 318 657 = 1584 + 594 = 1344 + 1334.

Leonhard Euler1707 – 1783

The smallest integerrepresented by x

4 + y4 in two

essentially di↵erent ways wasfound by Euler, it is635 318 657 =41⇥ 113⇥ 241⇥ 569.

[OEIS A216284] Number of solutions to the equationx4 + y

4 = n with x � y > 0.An infinite family with one parameter is known for non trivialsolutions to x

41 + x

42 = x

43 + x

44.

http://mathworld.wolfram.com/DiophantineEquation4thPowers.html

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Sums of k–th powers

One conjectures that given k � 5, if an integer can be writtenas xk + y

k, there is essentially a unique such representation.But there is no value of k for which this has been proved.

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Binary cyclotomic forms of higher degreeThe situation for binary cyclotomic forms is di↵erent when thedegree is 2 or when it is > 2 also for the following reason.A necessary and su�cient condition for a number m to berepresented by one of the quadratic forms �3, �4, is given bya congruence.By contrast, consider the quartic binary form�8(X, Y ) = X

4 + Y4. On the one hand, an odd integer

represented by �8 is of the form

N1,8(N3,8N5,8N7,8)4.

On the other hand, there are many integers of this form whichare not represented by �8.

[OEIS A004831] Numbers that are the sum of at most2 nonzero 4th powers.

0, 1, 2, 16, 17, 32, 81, 82, 97, 162, 256, 257, 272, 337, 512, 625, . . .

626, 641, 706, 881, 1250, 1296, 1297, 1312, 1377, 1552, 1921, 2401, 2402, 2417, 2482, 2592, 2657, 3026, 3697, 4096, 4097, 4112, 4177, 4352, 4721, 4802, 5392, 6497, 6561, 6562, 6577, 664230 / 47

Quartan primes

[OEIS A002645] Quartan primes: primes of the formx4 + y

4, x > 0, y > 0.

The list of prime numbers represented by �8 start with2, 17, 97, 257, 337, 641, 881, 1297, 2417, 2657, 3697, 4177,4721, 6577, 10657, 12401, 14657, 14897, 15937, 16561,28817, 38561, 39041, 49297, 54721, 65537, 65617, 66161,66977, 80177, 83537, 83777, 89041, 105601, 107377, 119617, . . .

It is not known whether this list is finite or not.

The largest known quartan prime is currently thelargest known generalized Fermat prime: The1 353 265-digit (145 31065 536)4 + 14.

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Primes of the form x2k + y

2k

[OEIS A002313] primes of the form x2 + y

2,[OEIS A002645] primes of the form x

4 + y4,

[OEIS A006686] primes of the form x8 + y

8,[OEIS A100266] primes of the form x

16 + y16,

[OEIS A100267] primes of the form x32 + y

32.

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Primes of the form X2 + Y

4

John Friedlander Henryk IwaniecHowever, it is known that there are infinitely many primenumbers of the form X

2 + Y4.

Friedlander, J. & Iwaniec, H. The polynomial X2 + Y

4

captures its primes, Ann. of Math. (2) 148 (1998), no. 3,945–1040.https://arxiv.org/pdf/math/9811185.pdf [A028916]

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K. Mahler (1933)

Let F be a binary form of degree d � 3 with nonzerodiscriminant.Denote by AF the area (Lebesgue measure) of the domain

{(x, y) 2 R2 | F (x, y) 1}.

For Z > 0 denote by NF (Z) the number of (x, y) 2 Z2 suchthat 0 < |F (x, y)| Z.Then

NF (Z) = AFZ2d +O(Z

1d�1 )

as Z ! 1.

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Kurt Mahler

Kurt Mahler1903 – 1988

Uber die mittlere Anzahl der Darstellungen grosser Zahlendurch binare Formen,Acta Math. 62 (1933), 91-166.https://carma.newcastle.edu.au/mahler/biography.html

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C.L. Stewart - S.Y. Xiao

Let F be a binary form of degree d � 3 with nonzerodiscriminant.There exists a positive constant CF > 0 such that the numberof integers of absolute value at most N which are representedby F (X, Y ) is asymptotic to CFN

2d +O(N�d) with �d <

2d ·

36 / 47

Cam Stewart and Stanley Yao Xiao

Cam Stewart Stanley Yao Xiao

C.L. Stewart and S. Yao Xiao, On the representation of

integers by binary forms,arXiv:1605.03427v2

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Cyclotomic binary forms of degree 4

(Joint work with Etienne Fouvry).

�5(X, Y ) = X4 +X

3Y +X

2Y

2 +XY3 + Y

4.

�8(X, Y ) = X4 + Y

4.

�12(X, Y ) = X4 �X

2Y

2 + Y4.

Also�10(X, Y ) = �5(X,�Y ) = X

4 �X3Y +X

2Y

2 �XY3 + Y

4.

For n 2 {5, 8, 12}, the number of positive integers m N

which can be written as m = �n(x, y) is asymptotic toC�nN

12 .

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Numbers represented by two cyclotomic binaryforms of degree 4

The number of integers N which are represented by two ofthe three quartic cyclotomic binary forms �5, �8 and �12 isbounded by O✏(N

38+✏).

Consequence : the number of integers N which arerepresented by a cyclotomic binary form of degree 4 isasymptotic to

C4N12 +O✏(N

38+✏),

whereC4 = C�5 + C�8 + C�12 .

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Numbers represented by a cyclotomic binary formof degree � d

Any prime number p is represented by a cyclotomic binaryform : �p(1, 1) = p.

Given an integer d � 2, we consider the set of positive integersm which can be written as m = �n(x, y) with n � d and(x, y) 2 Z2 satisfying max(|x|, |y|) � 2.

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Numbers represented by a cyclotomic binary formof degree � d

Let d � 6. The number of integers m N which can bewritten m = �n(x, y) with n � d and (x, y) 2 Z2 satisfyingmax(|x|, |y|) � 2 is asymptotic to

CdN2d +Od(N

2d+2 ),

withCd =

X

n

C�n ,

where the sum is over the set of integers n such that '(n) = d

and n is not congruent to 2 modulo 4.

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Isomorphic cyclotomic binary forms

Recall that the cyclotomic polynomials �n(t) 2 Z[t] satisfy�2n(t) = �n(�t) for odd n � 3.

For n1 and n2 positive integers with n1 < n2, the followingconditions are equivalent :(1) '(n1) = '(n2) and the two binary forms �n1 et �n2 areisomorphic.(2) The two binary forms �n1 and �n2 represent the sameintegers.(3) n1 is odd and n2 = 2n1.

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Even integers not represented by Euler totientfunction

The list of even integers which are not values of Euler 'function (i.e., for which Cd = 0) starts with

14, 26, 34, 38, 50, 62, 68,74,76, 86, 90, 94, 98, 114, 118,122,124, 134, 142, 146,152,154, 158, 170, 174, 182,186,188, 194, 202, 206, 214, 218, 230,234,236,242,244,246,248, 254, 258, 266, 274, 278,284,286,290, 298,302,304, 308, 314, 318, . . .

[OEIS A005277] Nontotients: even n such that'(m) = n has no solution.

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Numbers represented by two cyclotomic binaryforms of the same degree

Given two binary cyclotomic forms of the same degree and notisomorphic, and given ✏ > 0, for N ! 1 the number ofpositive integers N which are represented by these twoforms is bounded by

8><

>:

O✏(N3

dpd+✏) for d = 4, 6, 8,

Od,✏(N1d+✏) for d � 10.

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A weak but uniform bound

For d � 2 and N ! 1, the number of m N for whichthere exists n � d and (x, y) 2 Z2 with max(|x|, |y|) � 2 andm = �n(x, y) is bounded by

29N2d (logN)1.161.

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Further developments (work in progress)• Representation of integers by the binary forms Xn + Y

n,X

n � Yn and F n(X, Y ), where

F n(X, Y ) = Xn +X

n�1Y + · · ·+XY

n�1 + Yn.

• The sequence of Dickson polynomials of the first kind

(Dn)n�0 (resp. second kind (En)n�0) is defined by

Dn(X+Y ,XY ) = Xn+Y n (resp. En(X + Y ,XY ) = Fn(X,Y )).

Suggestion of Florian Luca : study the representation ofintegers by the polynomials Dn and En.• Cyclotomic Dickson polynomials. For n � 2,

n(X + Y ,XY ) = �n(X, Y ).

Study the representation of integers by the polynomials n.46 / 47

Universite Laval, Quebec July 9 – 13, 2018

Canadian Number Theory Association ConferenceCNTA XV

Generalization of the Landau–Ramanujan

constant for sums of two squaresto other binary forms

Michel Waldschmidt

Institut de Mathematiques de Jussieu — Paris VIhttp://webusers.imj-prg.fr/~michel.waldschmidt/

update: 03/07/2018

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